1. Sociology 690 – Data Analysis
Simple Quantitative
Data Analysis

2. Four Issues in Describing Quantity
1. Grouping/Graphing Quantitative Data
2. Describing Central Tendency
3. Describing Variation
4. Describing Co-variation

3. 1. Grouping Quantitative Data
If there are a large number of quantitative scores, one would not simply create a raw score frequency distribution, as that would contain too many unique scores and, therefore, not fulfill the data reduction goal.
Intervals and Real Limits
Widths and midpoints
Graphing grouped data

4. Grouping Data - Intervals
To group quantitative data, three rules are followed:
1. Make the intervals no greater than the most amount of information you are willing to lose.
2. Make the intervals in multiples of five.
3. Make the distribution intervals few enough to be internalized at a glance.

5. Grouping Data – Intervals Example
If these are the scores on a midterm:
{9,13,18,19,22,25,31,34,35,36,36,38,41,43,44,45}
The corresponding grouped frequency distribution would look like:
i fi
Total

6. Grouping Data - Real Limits
This implies the need for real limits as there are “gaps” in these intervals. The real limits of an interval are characterized by numbers that are plus and minus one-half unit on each side of stated limits:
For example:
the interval becomes 10.5 – 20.5
the interval 3.5 – 4.5 becomes 3.45 – 4.55

7. Grouped Data – Width and Midpoint
The width of an interval is simply the difference between the upper and lower real limits.
e.g  20.5 – 10.5 = 10
The midpoint is determined by calculating the interval width, dividing it by 2, and adding that number to the lower real limit.
e.g. 10/ = 15.5

8. Graphing Grouped Data
A Quantitative version of a bar graph is called an Histogram:
When the frequencies are connected via a line, it is call a frequency polygon:

9. 2. Describing Central Tendency
But we can do more than simply create a frequency distribution. We can also describe how these observations “bunch up” and how they “distribute”. Describing how they bunch up involves measures of
Modes
Medians
Means
Skew

10. Central Tendency - Modes
The mode for raw data is simply the most frequent score: e.g. {2,3,5,6,6,8}. The mode is 6.
The mode for grouped data is the midpoint of the interval containing the highest frequency (35.5 here):
i fi
Total

11. Central Tendency - Medians
The median for raw data is simply the score at the middle position. This involves taking the (N+1)/2 position and stating the associated value attached to it:
e.g. {2,3,5,6,8} (5+1)/2  the third position score
The third position score is 5.
e.g. {2,3,5,8} (4+1)/2  the 2.5 position score
The 2.5 position score is (3+5)/2 = 4

12. Medians for Grouped Data
The median for grouped data is:
For our previous distribution of scores,
the answer would be:
30.5 +((16/2-6)/6)*10
= = 33.83
i fi
Total

13. Central Tendency - Mean
For raw data, the mean is simply the sum of the values divided by N:
Suppose Xi = { 2,3,5,6}
The mean would be 16/4 = 4

14. Means for Grouped Data
For grouped data, the mean would be the sum of the frequencies times midpoints for each interval, that sum divided by N:
For our previous distribution,
the answer would be:
i fi
(5.5)+3(15.5)+2(25.5)+6(35.5)
(45.5) = 498 / 16 =
Total

15. 3. Describing Variation Range Mean Deviation Variance
Standard Scores (Z score)

16. Describing Variation - Range
The Range for raw scores is the highest minus the lowest score, plus one (i.e. inclusive)
The Range for grouped scores is the upper real limit of the highest interval minus the lower real limit of the lowest interval. In the case of our
previous distribution this would be
= 50
i fi
Total

17. Describing Variation – Mean Deviation
The mean deviation is the sum of all deviations, in absolute numbers, divided by N.
Consider the set of observations, {6,7,9,10} The mean is 8 and the MD is (|6-8|+|7-8|+|9-8|+|10-8|)/4 = 6/4 = 1.5

18. Mean Deviation for Grouped Data
Again grouped data implies we substitute frequencies and midpoints for values:
The mean would be $50,000 (satisfy yourself that that is true) and the MD would be (6|38-50|) + (8|43-50|) + (12|48-50|) + (12|53-50|) + (8|58-50|) +(4|63-50|) = = 304/50 = x 1000 = 6,080

19. Variation – The Variance
The variance for raw data is the sum of the squared deviations divided by N
Consider the set Xi { 6,7,9,10} The mean is 8 and the variance is ((6-8)2+(7-8)2+(9-8)2+(10-8)2)/4 = 2.5

20. Variance for Grouped Data
Frequencies and midpoints are still substituted for the values of Xi.
Again the mean is 50 and the Variance is 6(38-50)2 + 8(43-50)2 + 12(48-50) (53-50)2 + 8(58-50)2 + 4(63-60)2 = = 2690 / 50 = 53.8 x 1000 = $53,800. The Standard Deviation is the sq root of this.

21. 4. Covariance and Correlation
The Definition and Concept
The Formula
Proportional Reduction in Error and r2

22. Correlation – Definition and Concept
Visually we can observe the co-variation of two variables as a scatter diagram where the abscissa and ordinate are the quantitative continua and the points are simultaneously mapping of the pairs of scores.

23. Correlation - Formula
Think of the correlation as a proportional measure of the relationship between two variables. It consists of the co-variation divided by the average variation:

24. Correlation and P.R.E.
Consider this scatter diagram. The proportion of variation around the Y mean (variation before knowing X), less the proportion of variation around the regression line (variation after knowing x) is r2

25. Partial Correlation
IV. Quantitative Statistical Example of Elaboration
Step 1 – Construct the zero order Pearson’s correlations (r).
Assume rxy = .55 where x = divorce rates and y = suicide rates.
Further, assume that unemployment rates (z) is our control variable and that rxz = .60 and ryz = .40
.55 – (.6) (.4)
Step 2 – Calculate the partial correlation (rxy.z) = =
.42
Therefore, Z accounts for ( ) or 12% of Y and (.12/.30) or 40% of the relationship between X&Y
Before z (rxy)2 = .30
Step 3 – Draw conclusions
After z (rxy.z)2 = .18